1. If you were to receive $3 for every time a 7 or 11 was rolled with two fair dice and lost $1 for all the ther values, what is the expected value (mean) for you as a gambler in this experiment? (please show work so I know how to do it).

Question

1. If you were to receive $3 for every time a 7 or 11 was rolled with two fair dice and lost $1 for all the ther values, what is the expected value (mean) for you as a gambler in this experiment? (please show work so I know how to do it).

2. Going with the question above, would you continue to play this game if the payoff for a 7 or an 11 were raised to $4 instead of $3? Why or why not?

Answer 1

To find the expected value (mean) for a gambler in this experiment, we need to calculate the expected value for each possible outcome and then sum them up.

1. Expected Value Calculation (with $3 payout for 7 or 11):
Let's determine the probability of rolling a 7 or 11 first. There are 6 possible outcomes for each dice roll (numbers 1 to 6).

- Probability of rolling a 7:
There are six ways to roll a 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). So the probability of rolling a 7 is 6/36 or 1/6.

- Probability of rolling an 11:
There are two ways to roll an 11: (5, 6), (6, 5). So the probability of rolling an 11 is 2/36 or 1/18.

Now, let's calculate the expected value:

Expected value = (payout1 * probability1) + (payout2 * probability2) + ... + (payoutn * probabilityn)

In this case, the possible payouts are:
- $3 when rolling a 7 or 11
- -$1 for all other outcomes

So, the expected value can be calculated as follows:

Expected value = ($3 * 1/6) + ($3 * 1/18) + (-$1 * 10/36)
Expected value = $0.50

Therefore, the expected value for a gambler in this experiment, where the payout for a 7 or 11 is $3, is $0.50.

2. If the payout for a 7 or 11 were raised to $4 instead of $3, we would need to recalculate the expected value using the updated payout.

Expected value (with $4 payout for 7 or 11):

Expected value = ($4 * 1/6) + ($4 * 1/18) + (-$1 * 10/36)
Expected value = $0.67

Since the expected value is positive ($0.67), a gambler would continue to play this game if the payout for a 7 or 11 was raised to $4. This means that, on average, the gambler can expect to make a profit over the long run.

Note: It's important to consider other factors, such as the gambler's risk tolerance and the overall odds of winning, before making a decision to continue playing.